Draw a complicated diagram.
Include it several times in the document and don't optimise it. This one takes about 10s on my computer. Increasing the 89.9 in the first \foreach would mean it took even longer (though 89.99 produces an overflow). Drawing the same diagram several times would again increase the time it took.
\documentclass{article}
\thispagestyle{empty}
%\def\pgfsysdriver{pgfsys-tex4ht.def}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
% \x runs over the angles at which to draw the circles defining the
% torus
\foreach \x in {90,89.9,...,-90} { % change 89 to 80 or 45 for speed
% \elrad is the x-radius of the ellipse (technically, a circle seen
% from side on at angle \x). The 'max' is because at small angles
% then the real ellipse is too thin and the torus doesn't ``fill
% out'' nicely.
\pgfmathsetmacro\elrad{20*max(cos(\x),.1)}
% We draw the torus from the back to the front to get the right
% layering effect. To tint it, we define colours according to the
% angle, but need different colours for the left and right pieces.
% It'd be nice if the xcolor colour specification could take something
% computed by pdfmath, such as {red!\tint} but it doesn't appear to
% work, so we define the colours explicitly.
\pgfmathsetmacro\ltint{.9*abs(\x-45)/180}
\pgfmathsetmacro\rtint{.9*(1-abs(\x+45)/180)}
\definecolor{currentcolor}{rgb}{\ltint, 0, \ltint}
% This draws the right-hand circle.
\draw[color=currentcolor,fill=currentcolor] (xyz polar cs:angle=\x,y radius=.75,x radius=1.5) ellipse (\elrad pt and 20pt);
% This sets the colour correctly for the left-hand circle ...
\definecolor{currentcolor}{rgb}{\rtint, 0, \rtint}
% ... and draws it
\draw[color=currentcolor,fill=currentcolor] (xyz polar cs:angle=180-\x,radius=.75,x radius=1.5) ellipse (\elrad pt and 20pt);
% End of foreach statement
}
% Spheres are *much* easier!
\shadedraw[shading=ball,ball color=purple, white] (6.5,0) circle (1.5);
% As are the subsets of Euclidean space
\draw[fill=cyan] (-1,-4) rectangle (1,-3);
\draw[fill=cyan] (5.5,-4) rectangle (7.5,-3);
% The next three draw the maps, slightly curved for aesthetics.
\draw[->] (0,-2.8) .. controls (-.2,-2.2) .. (0,-1.6) node[pos=0.5, auto=left] {\(\psi\)};
\draw[->] (6.5,-1.6) .. controls (6.7,-2.2) .. (6.5,-2.8) node[pos=0.5, auto=left] {\(\phi^{-1}\)};
\draw[->] (2.5,0) .. controls (3.5,.2) .. (4.5,0) node[pos=0.5, auto=left] {\(f\)};
% Now we want to draw the codomains of the charts. Sticking cosines
% and sines directly into the coordinates doesn't seem to work so
% we define macros to hold the sines and cosines of the angles.
% \elrad is the angle on the torus at which to start.
\pgfmathsetmacro\elrad{cos(-135)}
% the circle drawn at the specific angle on the torus looks like an
% ellipse, \xrad and \yrad compute its major and minor semi-axes.
\pgfmathsetmacro\xrad{1.5cm-20pt*\elrad}
\pgfmathsetmacro\yrad{.75cm-20pt*sin(-135)}
% This draws the codomain of the chart on the torus.
\path[fill=cyan, fill opacity=.35] (xyz polar cs:angle=-135,radius=.75,x radius=1.5) ++(20pt*\elrad,0) arc (0:45:20*\elrad pt and 20pt) arc (-135:-45:\xrad pt and \yrad pt) arc (45:-45:-20*\elrad pt and 20pt) arc (-45:-135:\xrad pt and \yrad pt) arc (-45:0:20*\elrad pt and 20pt);
% Now we do the same for the sphere.
% We do this by drawing some great circles (aka ellipses) on the
% sphere and then ``clipping'' an overlaid (and slightly trans:parent)
% sphere by those great circles. Each great circle actually specifies
% one side of the ``clip'' so to make sure that the clip is big enough
% the arcs are completed by big rectangles (otherwise the clipping
% would join the end points directly).
\pgfmathsetmacro\tell{-sin(10)}
\pgfmathsetmacro\bell{sin(50)}
\pgfmathsetmacro\rell{1.5 * sin(50)}
\begin{scope}
\clip (6.5,0) +(-1.5,0) arc (-180:0:1.5 and 1.5*\tell) -- ++(0,-1.5) -- ++(-3,0) -- ++(0,1.5);
\clip (6.5,0) +(-1.5,0) arc (-180:0:1.5 and 1.5*\bell) -- ++(0,1.5) -- ++(-3,0) -- ++(0,-1.5);
\clip (6.5,0) +(0,1.5) arc (90:-90:\rell cm and 1.5 cm) -- ++(-1.5,0) -- ++(0,3) -- ++(1.5,0);
\clip (6.5,0) +(0,1.5) arc (90:-90:-\rell cm and 1.5 cm) -- ++(1.5,0) -- ++(0,3) -- ++(-1.5,0);
\fill[cyan, fill opacity=0.35] (6.5,0) circle (1.5);
\end{scope}
\end{tikzpicture}
\end{document}
Result:
Remarks:
- This is a diagram from a conference talk that I gave so it is not an example that I cooked up to answer this question.
- It is the torus that takes so long, it is drawn as a family of circles.
- When writing the seminar, I actually increased the step size considerably as it was taking so long to compile on each run, only decreasing it for the final compilation.
- When I gave this talk a second time, I found a quicker way of drawing the torus.
:)pdflatex thesis.tex 326.88s user 0.94s system 99% cpu 5:28.49 totalThat's 11 minutes every time I build that document. I pretty much already watched all of Youtube so this is not funny!;-)Anyway, I am actually a workaholic and this is merely a joke, although undoubtedly sometimes useful. I shall continue sharing this great site and passion for LaTeX with my acquaintances.:-)